How to scale all the edges proportionally
Alt+S command should do the job here. Don't know why it doesn't? What you can do is extrude the bottom edge loop downwards (E,Z), select the extruded face loop, press E,RMB, then Alt+S, S (to enable even thickness mode). Finally delete the unnecessary geometry.


Note: After doing this the edges lenghts are still not perfect. It's because the model you provided isn't perfect too.